Actually, Citytunneln will be operational already from the Europe-wide timetable change on December 12. Incredibly, they managed to finish a billion-euro infrastructure project six months early and under budget. ^

At Copenhagen main station and Copenhagen airport you can buy the ticket either from the Danish (DSB) or Swedish (Skånetrafiken) vending machines or the manual DSB ticket sales.

Make sure you get on a local Öresund train and not one of the SJ long-distance trains. Tickets for the latter you need to get from the SJ vending machine (yes there are three kinds!), but to Malmö it is often more expensive.

And finally, if you're going to central Malmö, the central station is the second stop after the bridge (third one when they have started using the tunnel). The first stop (Malmö Syd/South) is just a shopping center outside of town.